In this paper, we study nonlinear fractional (p,q)-difference equations equipped with separated nonlocal boundary conditions. The existence of solutions for the given problem is proven by applying Krasnoselskii’s fixed-point theorem and the Leray–Schauder alternative. In contrast, the uniqueness of the solutions is established by employing Banach’s contraction mapping principle. Examples illustrating the main results are also presented.

Fractional calculus is the field of mathematical analysis that investigates and applies integrals and derivatives of arbitrary order. Fractional q-calculus has been investigated and applied in a variety of research subjects including the fractional q-trapezoid and q-midpoint type inequalities. Fractional $(p,q)$ ( p , q ) -calculus on finite intervals, particularly the fractional $(p,q)$ ( p , q ) -integral inequalities, has been studied. In this paper, we study two identities for continuous functions in the form of fractional $(p,q)$ ( p , q ) -integral on finite intervals. Then, the obtained results are used to derive some fractional $(p,q)$ ( p , q ) -trapezoid and $(p,q)$ ( p , q ) -midpoint type inequalities.

Fractional q-calculus has been investigated and applied in a variety of fields in mathematical areas including fractional q-integral inequalities. In this paper, we study fractional (p,q)-calculus on finite intervals and give some basic properties. In particular, some fractional (p,q)-integral inequalities on finite intervals are proven.

In this paper, we study the existence of solutions to a fractional (p, q)-difference equation equipped with separate local boundary value conditions. The uniqueness of solutions is established by means of Banach’s contraction mapping principle, while the existence results of solutions are obtained by applying Krasnoselskii’s fixed-point theorem and the Leary–Schauder alternative. Some examples illustrating the main results are also presented.